Skip to main content
SpringerLink
Log in

Strongly regular graphs with Hoffman’s condition

  • Published:
Proceedings of the Steklov Institute of Mathematics Aims and scope Submit manuscript

Abstract

It is known that if the minimal eigenvalue of a graph is −2, then the graph satisfies Hoffman’s condition; i.e., for any generated complete bipartite subgraph K 1,3 with parts {p} and {q 1, q 2, q 3}, any vertex distinct from p and adjacent to two vertices from the second part is not adjacent to the third vertex and is adjacent to p. We prove the converse statement, formulated for strongly regular graphs containing a 3-claw and satisfying the condition gm > 1.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. V. V. Kabanov and A. A. Makhnev, Izv. Ural’sk. Gos. Univ. 10, 44 (1998).

    MathSciNet  Google Scholar 

  2. V. V. Kabanov, A. A. Makhnev, and D. V. Paduchikh, Mat. Zametki 74, 396 (2003).

    MathSciNet  Google Scholar 

  3. A. E. Brouwer, A. M. Cohen, and A. Neumaier, Distance-Regular Graphs (Springer-Verlag, Berlin, 1989).

    MATH  Google Scholar 

  4. A. E. Brouwer and J. H. van Lint, in Enumeration and Design, Ed. by D. M. Jackson and S. A. Vanstone (Academic, New York, 1984), pp. 85–122.

    Google Scholar 

  5. L. C. Chang, Sci. Record 3, 604 (1949).

    Google Scholar 

  6. L. C. Chang, Sci. Record 4, 12 (1950).

    Google Scholar 

  7. M. D. Hestenes and D. G. Higman, SIAM-AMS Proc. 4, 141 (1971).

    MathSciNet  Google Scholar 

  8. A. J. Hoffman, Ann. Math. Statist. 31, 492 (1960).

    Article  MathSciNet  MATH  Google Scholar 

  9. A. J. Hoffman, IBM J. Res. Develop. 4, 487 (1960).

    Article  MATH  MathSciNet  Google Scholar 

  10. J. J. Seidel, Linear Algebra Appl. 1, 281 (1968).

    Article  MATH  MathSciNet  Google Scholar 

  11. S. S. Shrikhande, Ann. Math. Statist. 30, 781 (1959).

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, ul. S. Kovalevskoi 16, Yekaterinburg, 620219, Russia

    V. V. Kabanov & S. V. Unegov

Authors
  1. V. V. Kabanov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. V. Unegov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. V. Kabanov.

Additional information

Original Russian Text © V.V. Kabanov, S.V. Unegov, 2007, published in Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2007, Vol. 13, No. 3.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kabanov, V.V., Unegov, S.V. Strongly regular graphs with Hoffman’s condition. Proc. Steklov Inst. Math. 261 (Suppl 1), 107–112 (2008). https://doi.org/10.1134/S008154380805009X

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S008154380805009X

Keywords

Search

Navigation